arXiv:1202.3908v3 [cond-mat.dis-nn] 28 Aug 2012 lwdnmc nsldHe solid in ult dynamics how observed slow experiment in recent role A a forms[12]. play supersolid may the disorder exper that Strong suggest evidence systems. mental bosonic in disorder betwee connection and the superfluidity on interests immense spurred has [11] supersolid [11]. of hints (SS) formation the us of give mechanism may microscopic the structure on amorphous granular frozen of a behavior in the perfluidity understanding particular, [8– In is fluctuations quantum ask 10]. by changed to be temperature question can transition the natural the how in One phase diagram[5–7]. glass phase spin disorder ther the and of bonds, FM reentrance of exists fraction the (decreases) increases one oantc(M n niermgei AM od with bonds (AFM) antiferromagnetic probability and f (NN) spins (FM) nearest-neighbor Ising romagnetic distributed which randomly in via model[2], interact (EA) Edwards-Anderson or re rniinfo Gt M(F)paea critical a second at a phase has (AFM) FM model a this to that concentration SG shown a been from has transition order It ex- dimens 4]. three and [3, in transition (3D) simulations glass temperature Carlo finite a Monte hibits with extensively studied o sn neatos hc ast netne hardcore extended an to maps un- which left interactions, yet Ising are dom SS/SuG this a na In into the transitions detail, explored. various some the in of studied ture been have states solid/glassy state[9]. superglass the ar stabilizing in frustration sup crucial random with both and coexist fluctuations can quantum glassiness and fluidity, that 3D a indicate on lattice interactions cubic Bose- frustrating hardcore random extended with the model Quantum Hubbard on 15]. studies 14, (QMC) 10, [9, Carlo Monte (SuG) “superglass” a or supersolid, htehbt pngas(G eairi the is behavior (SG) glass mod spin simplest exhibits the ex- that Traditionally, with phases dynamics[1]. glassy slow hosting tremely disorder, quenched with tems h bevto fsproiiyi oi eim4(He 4 Helium solid in supersolidity of observation The hl h hs rniin nosprudt rquantum or superfluidity into transitions phase the While Introduction– p cin natredmninlcbcltie nteclassic the In lattice. cubic dimensional three a on actions sn oe ihconcentration with model Ising clconcentration ical iieteceitneo uefliiyadgas re “s ( order to glassy transition and superfluidity of coexistence the bilize h rniint h lsypaeocr tahigher a at occurs phase glassy the to transition the ASnmes 55.k 54.g 05.50.+q 75.40.Mg, 75.50.Lk, numbers: PACS and p c esuyteitrlyo uefliiy lsyadmagnetic and glassy superfluidity, of interplay the study We Letter = pngassaefutae antcsys- magnetic frustrated are glasses Spin 1 − p c cl p esuyteXZmdlwt ran- with model XXZ the study we , p ∼ epciey hsmdlhsbeen has model This respectively. c 4 ainlTia nvriy o ,Sc ,RoeetR. Ta Rd., Roosevelt 4, Sec. 1, No. University, Taiwan National ainlTia nvriy o ,Sc ,RoeetR. Ta Rd., Roosevelt 4, Sec. 1, No. University, Taiwan National p > 1] ugsigagas yeof type glassy a suggesting [13], 0 . p 7( 77 c cl c cl ncnrs,atfroantcodrceit ihsuper with coexists order antiferromagnetic contrast, In . ∼ p c 0 . 77 1 = 2 u unu ot al iuainrslsso htquant that show results simulation Carlo Monte quantum Our . etrfrAvne td nTertclScience, Theoretical in Study Advanced for Center uigtedsre nsuperglasses in disorder the Tuning p − ffroantcbns hc ot glassy-ferromagneti a hosts which bonds, ferromagnetic of ee Larson Derek ± p J c cl sn model, Ising ∼ Dtd oebr1,2018) 11, November (Dated: 1 0 eateto Physics, of Department . 23) ions 1 su- ra- er- er- as 4 el i- p n igJrKao Ying-Jer and n e e s ) - - . llmt hsmdlrdcst a to reduces model this limit, al prls”,adsitte(super)glassy-ferromagnetic the shift and uperglass”), where lsia Amdlif model EA classical ntesi oe orsodt unu oi hsswith phases solid quantum to correspond (0 model spin the in M F n G ihcrepnigodrparameters: order corresponding with are orders SG, long-range magnetization and diagonal AFM Possible FM, model. this in ders simplicity. for language spin the use will we following, aaee and parameter ie as given yatasomto ftebsncoeaost pn12ope spin-1/2 to operators ators: bosonic the of transformation a By prtrfrhr-oebsn tltiesite lattice at bosons hard-core for operator where o Nitrcinis interaction NN dom n u-Mtastosarises. SuG-S transitions for SuG-FM asymmetry and an and concentration, critical higher a o eprtr hs onayt ihrciia concen- critical higher a than to tration boundary the phase tem- pushing temperature by the low transition increases SuG-FM the greatly of terms dependence pre perature exchange the that quantum demonstrate of We ence interactions. the di in of present amount the der tuning by transitions characterize to interactions, order frustrating random with model Bose-Hubbard eraiympe notesadr X model: XXZ standard the into mapped readily be H , hr xs ohdaoa n f-ignlln-ag or- long-range off-diagonal and diagonal both exist There Model– H 0 resi h X oe ihrno sn inter- Ising random with model XXZ the in orders = , 0) = S − h J i ,j i, z p xy − and h X ( = i,j V h i X i,j ij 2 = p i n h aitna o adoebsnwt ran- a with boson hardcore for Hamiltonian The ,2, 1, niae h Nltiesites. lattice NN the indicates c cl V ( = ) i i ,π π π, π, V ij m h u-Spaebudr sas rw to drawn also is boundary phase SuG-SS The . J − t ∗ ij ( z n = S 1 pδ and udt ofr uesld and supersolid, a form to fluidity r neatoswt ioa distribution bimodal a with interactions are pi16 Taiwan 106, ipei pi16 Taiwan 106, ipei i / i z − ( 2 S N 1 ) V S , 1 j z J ij reigvcos epciey nthe In respectively. vectors, ordering J P / z − i − 2)( xy − i = ± 2 1 = S V n 0 = J J i z j V b (1 + ) xy rniina crit- a at transition c mflcutossta- fluctuations um − Edwards-Anderson i F) tgee magnetization staggered (FM), ij and , 1 h hsmdlrdcst the to reduces model This . h MadAMphases AFM and FM The . X i,j / 2) i ( − S − S i + i − p t ) h X S = δ i,j j + ( V b i i n ( , + i † ij b hsmdlcan model this , t i i † S + stehopping the is b stenumber the is j i + + V S ) b j − . i b ) j † , ) , sor- (3) (2) (1) in s- r- S 2

1 i z 1.5 ms = N i(−1) Si (AFM), and the Edwards-Anderson or- der parameter (SG), defined as PM P

1 z 2 qEA = hSi i , (4) N 1 SG " i # AFM X av SuG FM where h· · · i denotes a thermal average and [··· ]av an aver- T age over disorder realizations. As the EA order parameter will also capture FM and AFM ordering, one must look for a 0.5 non-zero qEA while the other order parameters remain zero in order to identify an SG phase. To determine the phase transition point, we look at the SS Binder cumulants [16] for the order parameter, which should 0 cross at the transition point for different system sizes. How- 0 0.2 0.4 0.6 0.8 1 ever, the existence of corrections to scaling cause pairs of p small sizes to intersect away from the true critical point. To FIG. 1. (Color online) T vs p phase diagram. The dashed lines improve the accuracy of the measurement, we look at the se- indicate the position of the classical transitions SG-(A)FM, and the ries of intersection points created by successive pairs of sizes solid lines with arrows show the trend of the phase boundary shifts L and L +1 which should converge to the exact value as a we find in the quantum model. The blue circles are our estimates power law with the exponent determined by the leading cor- for the superfluid transition, with the approximate phase boundary rection to scaling. To get statistical estimates of these cross- drawn as the blue arc. Lastly, the red arc denotes the PM-SG phase ing points, we perform a bootstrap resampling on the raw data boundary. (Nboot = 100), fitting a 2nd-order polynomial to each size, and calculating the resulting intersection. The Binder cumu- TABLE I. Parameters of the simulations lants of the magnetization and staggered magnetization are β L p range ∼ Nsamp ∼ Nsweep 4 4 0.70 6 0.20 - 0.26 600 8000 1 hm i av 1 hmsi av gm = 3 − ; gsm = 3 − 0.70 8 0.20 - 0.26 400 16000 2 2 2 2 2 2 [hm i]av ! [hmsi]av ! 0.70 10 0.20 - 0.26 500 64000 (5) 0.70 12 0.20 - 0.26 250 128000 which approach one in the FM/AFM phase and goes to zero 1.00 3 0.70 - 0.80 2000 4000 otherwise. We measure the superfluid density through wind- 1.00 4 0.75 - 0.78 2000 16000 ing number fluctuations[17], 1.00 5 0.75 - 0.78 800 128000 1.00 6 0.75 - 0.78 150 256000 1.00 6 0.12 - 0.28 1000 16000 1 2 1.00 8 0.12 - 0.28 400 16000 ρs = hW i , (6) 3βNt  k  1.00 10 0.12 - 0.28 400 64000 k=x,y,z X av   2.00 6 0.20 - 0.28 500 8000 where Wk is the winding number along the k direction. In 2.00 8 0.20 - 0.28 500 128000 our simulation, we identify the SS phase by the coexistence SSEa 3 0.75 - 0.79 4700 16000 of AFM and SF orders, and the SuG phase by the coexistence SSE 4 0.75 - 0.79 4000 16000 of SG and SF orders. SSE 5 0.75 - 0.79 4000 16000 Method– We use the Worm Algorithm QMC [18] to SSE 6 0.75 - 0.79 4000 32000 study this model, as well as the Stochastic Series Expan- a Various temperatures are simulated simultaneously in the parallel sion (SSE) [19] with the parallel tempering [20] near the tempering. FM boundary to access low temperatures[21]. We simulate many different realizations of the disorder–sets of interactions Vij –for a given parameter set. In the following, we choose phases: worm update loopstend to be long while in SF phases, V/t = 4 in order to maximize the extent of the superglass and under increasing ferromagnetic order the worms have a phase at p = 0.5[9]. Each realization is simulated indepen- low probability of hopping producing short loops. This ef- dently and equilibrium is determined by reaching measure- fect is significantly reduced in our SSE implementations since ments that stabilize within our error bars. Table I contains each MC step has an adaptively determined number of opera- the details of our simulation parameters. For our worm al- tor loop updates. gorithm implementation a Monte Carlo (MC) step is defined Results– Figure 1 summarizes our QMC results in a 3 as Nsites = L completed worm loop updates, while for temperature-disorder (T -p) phase diagram. At high temper- SSE we define it the same as Ref. [19] with the addition of atures, we expect the model behaves classically, and is de- one replica exchange sweep. Equilibration times vary across scribed by the 3D EA model. The classical phase diagram is 3

1.1 0.785 (a) 0.78 1 0.775 β 0.77 0.9 =1.0 0.765 m (L)

0.8 c 0.76 g p β=3.0 0.755 0.7 0.75 L=4 β = 3.0 0.745 0.6 L=5 β = 2.0 L=6 0.74 β = 1.0 (b) 0.5 0.735 0.75 0.76 0.77 0.78 0.79 0.8 3 3.5 4 4.5 5 p L for (L,L+1) crossing

FIG. 2. (Color online) (a) Finite size scaling of gm at β = 1 and 3 to determine the SG-FM transition. Data for β = 1 are shifted vertically for cl clarity. Shaded areas indicate the crossing for different sizes. Dotted line shows the position of the zero-temperature classical transition (pc ). Our pc estimates show stronger reentrance. (b) Results of bootstrap estimates for the crossing points for adjacent sizes. Dotted line shows the position of the zero-temperature classical transition. Deviation from the classical transition point is clearly seen in the large size data points at β = 2 and 3.

from the classical behavior. Likely this is due to the encroach- 1.8 L= 6 ing superfluid order and subsequent competition. Meanwhile, β=0.7 1.6 L= 8 the AFM state allows for superfluidity, creating a region of L=10 supersolidity within the arm of the superfluid transition line, 1.4 L=12 β=1.0 which may be why the AFM state is more stable in the quan- 1.2 tum model. Furthermore, the SuG-SS line pushes deeper into the SG phase, suggesting interesting correlation between the sm β

g =2.0 1 AFM and SF orders. 0.8 The results for our analysis of the magnetic ordering are 0.6 presented in Fig. 2. Already at β = 1 we see evidence for a shift in the phase boundary. By β =3 in Fig. 2 (a), it is clear 0.4 there would not be a transition for p < 0.78, given the posi- 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 tion of the data crossings as shown. This is markedly different cl p than the classical results. The classical transition line pc as a function of β is only weakly temperature dependent, shifting less than a percent between the multicritical point and zero FIG. 3. (Color online) Finite size scaling of the staggered magne- temperature[7]. Here, that shift is at least 4 times as large. tization cumulant gsm. Data for β = 1 and β = 0.7 are shifted We have studied the superfluid transition as well for these pa- vertically for clarity. Results at β = 0.7 are in line with classical be- rameters, and our estimates (see Fig. 1) are only rough due to havior while β = 1, 2 suggest the SuG-SS transition has been drawn the non-trivial nature of the superfluid scaling form. For a 3D into the classical SG region. α XY model, one expects ρs ∼ 1/L with α = 1 away from quantum criticality. However, we find α > 1 as the model symmetric about p = 0.5 if one identifies the FM with the enters the glass phase with, for example, α ≈ 1.4 − 1.8 for AFM regime. The classical 3D EA model undergoes a T =0 the transition at p =0.5 [24]. cl SG-FM transition at pc ∼ 0.770, and shows slight reen- The specific mechanism behind the increased temperature trant behavior at finite T (red dashed lines). It is suggested dependenceof the SuG-FM transition should be rooted in how that the finite-temperature SG-FM transition belongs to a new superfluidity favors the spin glass phase over the ferromag- universality class with critical exponents ν = 0.96(2) and netic phase. One can imagine that clusters of FM-ordered η = −0.39(2) [7]. The SG-FM, PM-SG and PM-FM tran- spins are suppressed as the system can gain kinetic energy by sition lines meet at a multi-critical point p = p∗ (not shown), destroying local ferromagnetic order and allowing particles to which sits on the Nishimori line[7, 22], and the PM-SG tran- hop. On the other hand, glassy clusters more readily allow sition for 1 − p∗

getically neutral, and this would turn the local AFM order into FM FM order. However, after this move the system is now con- AFM siderably more constrained regarding the number of hoppings available. Originally, the two interior sites could be involved with seven different hop moves, but these sites now have a single move: to return back to the original state. In this way, the AFM state is favored as it allows for more gain in kinetic energy. Summary– We have presented our QMC results of a model exhibiting a superglass phase in order to study the phase transitions achieved by directly tuning the level of dis- order present. These results indicate that the addition of the FIG. 4. A typical 2D snapshot near the SuG-SS line. The hop exchange terms act to stabilize the classical spin glass phase move indicated is energetically neutral with respect to the bonds and against the formation of ferromagnetic clusters which impede destroys the local AFM order. However, the new state allows fewer hopping. In addition, the favoring of AFM bondsto FM bonds fluctuations so is not favored with respect to entropy. leads to a shift in the SuG-SS phase boundary. An interesting issue not addressed here is whether there is always a SG phase between the FM and SuG phases. Recent work[25] on disor- We also looked at the behavior on the other side, p < 0.5, dered Bose systems suggests that this may be true. Unfortu- which is different due to the asymmetry of the ground state nately, our current precision is not high enough to be conclu- against quantum fluctuations. Notably, we find superfluid sive. Future work is required to clarify this issue. transitions at higher temperatures: βc(1 − p) < βc(p). Fig- We thank Anders Sandvik and Markus M¨uller for helpful ure 3 shows data for the Binder cumulant using the staggered discussions, and P. C. Chen for the use of his cluster at NTHU. magnetization, gsm at β = 1 and 2. The staggered magneti- We are grateful to National Center for High-Performance zation data suffer from larger FSE and subsequently require Computing, and Computer and Information Networking Cen- larger system sizes to achieve comparable precision, restrict- ter at NTU for the support of high-performancecomputing fa- ing the current study from exploring lower temperatures. cilities. This work was partly supported by the NSC in Taiwan The position of the crossing at β =1 lies around pc ∼ 0.25 through Grants No. 100-2112-M-002 -013 -MY3, 100-2120- indicating that the SuG-SS line is noticeably shifted from clas- M-002-00 (Y.J.K.), and by NTU Grant numbers 10R80909-4 sical behavior. The data at lower T are less conclusive due to (Y.J.K.). Travel support from National Center for Theoretical larger sizes being out of reach at present, but suggest an even Sciences is also acknowledged. larger shift. Above the SF transition at β = 0.7(T = 1.43), the AFM-SG transition agrees with the classical transition cl SUPPLEMENTARY MATERIAL point p = 1 − pc = 0.23. Thus, the stronger the quantum mechanical nature becomes–the deeper into the superfluid be- havior we go–the stronger the AFM order appears. This is at Figure 5 shows the equilibration data of the magnetic first counterintuitive, but we can provide a simple, consistent Binder cumulant, using the stochastic series expansion (SSE) picture for both this and the SuG-FM behavior. When an FM quantum Monte Carlo with parallel tempering (PT), near the bond is satisfied, i.e. two neighboring sites are in the same ferromagnetic transition for L = 6. Each data point is ob- state, this forbids hopping along that bond. Conversely, when tained by first equilibrating for a given number of sweeps an AFM bond is satisfied, the two sites are in an opposite state (Nsw), and the perform measurements for the same number of 10 and will allow the system to gain kinetic energy through hops. sweeps. For example, the point at Nsw = 2 will have1024 As this is energetically favorable, we can consider the AFM sweeps for equilibration and then 1024 sweeps of measure- bonds to have a larger effective bond strength. Consequently, ment. In addition, averaging over different disorder realiza- the fraction of frustrating FM bonds it would require to de- tions is carried out. It is clear that the simulations are well- stroy the AFM order should rise, and we see the present shift equilibrated in the doping range of interests, p =0.75 − 0.79, in phase boundary. and at all temperatures T = 0.3333, 0.5000, 1.0000. Smaller Speaking more generally, we can consider the change in the sizes are all strongly equilibrated. entropy of the system, with respect to quantum fluctuations, when moving towards higher AFM or FM order. Under pure FM order no hopping is allowed, while under pure AFM order each site has the ability to engage in virtual hopping. For the ∗ [email protected] illustration purpose, Fig. 4 shows a typical local 2D snapshot [1] K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986). of a system with AFM order being frustrated by FM bonds. [2] S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5, 965 The indicated move would go against three NN bonds while (1975). satisfying three others, making the diagonal component ener- [3] N. Kawashima and A. P. Young, Phys. Rev. B 53, R484 (1996). 5

[4] H. G. Ballesteros, A. Cruz, L. A. Fern´andez, V. Mart´ın-Mayor, [15] J. Wu and P. Phillips, Phys. Rev. B 78, 014515 (2008). J. Pech, J. J. Ruiz-Lorenzo, A. Taranc´on, P. T´ellez, C. L. Ullod, [16] K. Binder, Z. Phys. B 43, 119 (1981). and C. Ungil, Phys. Rev. B 62, 14237 (2000). [17] E. L. Pollock and D. M. Ceperley, [5] A. K. Hartmann, Phys. Rev. B 59, 3617 (1999). Phys. Rev. B 36, 8343 (1987). [6] M. Hasenbusch, A. Pelissetto, and E. Vicari, Phys. Rev. B 78, [18] N. Prokof’ev, B. Svistunov, and I. Tupit- 214205 (2008). syn, Physics Letters A 238, 253 (1998); [7] G. Ceccarelli, A. Pelissetto, and E. Vicari, Journal of Experimental and Theoretical Physics 87, 310 (1998), Phys. Rev. B 84, 134202 (2011). 10.1134/1.558661. [8] T. E. Markland, J. A. Morrone, B. J. Berne, K. Miyazaki, E. Ra- [19] A. W. Sandvik, Phys. Rev. B 59, R14157 (1999). bani, and D. R. Reichman, Nat Phys 7, 134 (2011). [20] K. Hukushima and K. Nemoto, J. Phys. [9] K.-M. Tam et al., Phys. Rev. Lett. 104, 215301 (2010). Soc. Jpn. 65, 1604 (1996); R. G. Melko, [10] X. Yu and M. M¨uller, Phys. Rev. B 85, 104205 (2012). Journal of Physics: Condensed Matter 19, 145203 (2007). [11] E. Kim and M. H. W. Chan, Nature 427, 225 (2004); [21] See Supplemental Material at [URL will be inserted by pub- Science 305, 1941 (2004). lisher] for the equilibration data near the FM phase boundary. [12] A. S. C. Rittner and J. D. Reppy, [22] H. Nishimori, Prog. Theor. Phys. 66, 1169 (1981). Phys. Rev. Lett. 98, 175302 (2007); [23] M. Hasenbusch, F. P. Toldin, A. Pelissetto, and E. Vicari, Phys. Rev. Lett. 97, 165301 (2006). Phys. Rev. B 76, 094402 (2007). [13] B. Hunt, E. Pratee, V. Gadagkar, M. Yamashita, A. Balatsky, [24] D. A. Larson and Y.-J. Kao, Unpublished (2012). and J. C. Davis, Science 324, 632 (2009). [25] L. Pollet, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, [14] G. Biroli, C. Chamon, and F. Zamponi, Phys. Rev. Lett. 103, 140402 (2009). Phys. Rev. B 78, 224306 (2008). 6

Equilibration: PTSSE L=6 p=0.75 Equilibration: PTSSE L=6 p=0.76 0.6 0.69

0.59 0.68

0.58 0.67 0.66 0.57 0.65 0.56 0.64 0.55 0.63 0.54 0.62 0.53 0.61 Binder Ration of Magnetization Binder Ration of Magnetization

0.52 0.6

0.51 T=0.3333 T=0.3333 T=0.5000 0.59 T=0.5000 T=1.0000 T=1.0000 0.5 0.58 11 11.5 12 12.5 13 13.5 14 14.5 15 11 11.5 12 12.5 13 13.5 14 14.5 15

Log2(Nsw) Log2(Nsw) Equilibration: PTSSE L=6 p=0.77 Equilibration: PTSSE L=6 p=0.78 0.77 0.84

0.76 0.83

0.75 0.82

0.74 0.81 0.73 0.8 0.72 0.79 0.71

Binder Ration of Magnetization Binder Ration of Magnetization 0.78 0.7

0.69 T=0.3333 0.77 T=0.3333 T=0.5000 T=0.5000 T=1.0000 T=1.0000 0.68 0.76 11 12 13 14 15 16 11 11.5 12 12.5 13 13.5 14 14.5 15

Log2(Nsw) Log2(Nsw) Equilibration: PTSSE L=6 p=0.79 0.9

0.89

0.88

0.87

0.86

0.85 Binder Ration of Magnetization

0.84 T=0.3333 T=0.5000 T=1.0000 0.83 11 11.5 12 12.5 13 13.5 14 14.5 15

Log2(Nsw)

FIG. 5. (Color online) Equilibration data for the SSE at the largest system size (L = 6) for p = 0.75 − 0.79. Smaller sizes are all more strongly equilibrated.